Research activity
The IsoGeometric Method (IGM) stands for class of discretisation
techniques for partial differential equations (PDEs) that
addresses the interoperability between Computer
Aided Design (CAD) and numerical simulation of PDEs. CAD software, used in industry for geometric modeling, typically describes physical domains by means of Non-Uniform Rational B-Splines (NURBS) and the interface between CAD output and classical numerical schemes calls for expensive re-meshing methods that result in approximate representation of domains. IGMs are NURBS-based schemes for solving PDEs whose benefits go beyond the improved interoperability with CAD. Indeed, they provide a substantial increase of the accuracy-to-computational-effort ratio and, thanks to the use of high-degree smooth NURBS within the numerical scheme, they outperform classical numerical schemes in most academic benchmarks. However, the mathematical understanding of the IGM is still incomplete and likely we are far from exploiting its full potential. The use of higher-degree IGM for real-world applications asks for new tools allowing for the efficient construction and solution of the linear system, time integration, flexible local mesh refinement, and so on. This research activity is aimed at providing the crucial knowledge to further develop the IGM into a highly accurate and stable methodology, having an impact in the field of numerical simulation, particularly when accuracy is essential both in geometry and fields representation.
In particular, the following topics are currently under investigation:
- theory of spline/NURBS spaces (high-degree splines, hierarchical splines, T-splines)
- design of spline spaces for complex geometries (multipatch, trimming, software interface to solid modelers)
- stability and well-posedness of IGMs for various class of applications: solid mechanics, contact problems, fluid dynamics, electromagnetism (De Rahm compatible splines)
- linear solvers for the IGM
- adaptive method for IGM based on hierarchical splines and T-splines
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